It could be “open season on pedestrians,” warns a lawyer for a woman run down by a bike-riding food deliveryman — whose employer’s insurer claims it isn’t responsible for anything that happens outside the restaurant.

“It’s fair game to hit the public,” warned lawyer Justin Blitz, whose client, Mary Burke, was left with a fractured skull by an allegedly out-of-control bike deliverer.

The insurer of the Gramercy restaurant Red House claims it shouldn’t have to pay the woman a red cent for the frightening accident, which occurred when she was six months pregnant, because it didn’t occur in the eatery itself.

“It’s the wild, wild West out there,” cautioned Blitz of the delivery people, who often ride on the sidewalk and the wrong way on streets and in bike lanes.

Blitz said the argument by Leading Insurance Services, if successful in court, would give a free pass to insurers of city restaurants.

Burke, 35, was walking up Sixth Avenue with a friend last Dec. 5 and crossing West 16th Street with the light, in a crosswalk, when she was bowled over by a deliveryman on a bike, she charges in her lawsuit.

“Next thing I knew, I was on the ground, and I had passed out or blacked out for the moment,” Burke said. “The delivery guy . . . had no helmet on. He was going in the wrong direction.”

Burke was taken to the hospital, where “I ended up having a skull fracture and bleeding on the brain . . . The baby was fine,” she said.

She missed several weeks at her job with Thomson Reuters, where she works in business management.

She later delivered a healthy baby boy.

In September, Burke filed a lawsuit in Manhattan Supreme Court against the deliveryman, identified only as John Doe, and his employer, Red House restaurant on East 14th Street.

Her lawyer, Blitz, said the insurer originally offered a mere $5,000 to settle the case, which he laughed off as ridiculously low.

Last month a lawyer for Leading Insurance Services wrote a blunt letter to the eatery, saying that the insurance policy limited coverage to injuries or damages that occurred on the “premises.”

Leading Insurance did not respond to a request for comment. A lawyer for Red House did not return a call requesting comment.